Given #f(x)=x^2-9#
#{:
(color(white)("X")x, color(white)("XXX"), f(x)),
(-3, color(white)("XXX"), color(white)("X")0),
(-2, color(white)("XXX"),-5),
(-1, color(white)("XXX"), -8),
( color(white)("X")0, color(white)("XXX"), -9),
(color(white)("X") 1, color(white)("XXX"), -8),
(color(white)("X") 2, color(white)("XXX"), -5),
( color(white)("X")3, color(white)("XXX"), color(white)("X")0)
:}#
Since the width of each trapezoid is 1, the area will be equal to the average height.
#{:
("trapezoid",
color(white)("XX")"from x =",
color(white)("XX")"to x=",
color(white)("XX")"average height",
color(white)("XX")"area"
),
(" ",
color(white)("XX")-3,
color(white)("XX")-2,
color(white)("XX")(0+(-5))/2=-2.5,
color(white)("XX")-2.5
),
(" ",
color(white)("XX")-2,
color(white)("XX")-1,
color(white)("XX")((-5)+(-8))/2=-6.5,
color(white)("XX")-6.5
),
(" ",
color(white)("XX")-1,
color(white)("XXX")0,
color(white)("XX")((-8)+(-9))/2=-8.5,
color(white)("XX")-8.5
),
(" ",
color(white)("XXX")0,
color(white)("XXX")1,
color(white)("XX")((-9)+(-8))/2=-8.5,
color(white)("XX")-8.5
),
(" ",
color(white)("XXX")1,
color(white)("XXX")2,
color(white)("XX")((-8)+(-5))/2=-6.5,
color(white)("XX")-6.5
),
(" ",
color(white)("XXX")2,
color(white)("XXX")3,
color(white)("XX")((-5)+0)/2=-2.5,
color(white)("XX")-2.5
)
:}#
Total area of trapezoids
#= (-2.5)+(-6.5)+(-8.5)+(-8.5)+(-6.5)+(-2.5)#
#= -55#